Monday, November 1, 2010

Space Thickness, Supermanifold and Multidimensional Time

We used to conceptualize the geometry elements such as point, line, surface and space as having, respectively, zeroed, one, two and three dimensions. There is nothing wrong with that as far as we are dealing with abstract objects such as a corner point between a floor and two walls, meeting line between ceiling and wall, table's surface or hall's spaciousness.

However, we cannot apply such a concept for real bodies, whatever the size is. A grain of sand is not a zeroed-dimensional object, but a three-dimensional cubic-like body, which has small length, width, and thickness. A string is a three-dimensional long cylindrical object having a small section. Similarly, a piece of paper is a three-dimensional surface object whose thickness is very thin (Figure-1). Had their thickness been reduced to zero, those objects would all have gone into thin air.

Nature does not seem to give any exception to natural bodies such as space or any other higher-dimensional spacetimes. For their existence to have physical meaning, all those bodies should have thickness.

It implies that space or spacetime, whatever its dimensions, are always be embedded in an ambient spacetime of at least one dimension higher. At the same token, the later is also embedded in turn in another much higher manifold (Figure-2).  This kind of infinite regress makes us believe that nature is vast and infinite, not only its areas but also its dimensions. 

System and Surroundings

Now, the formulation of the laws of nature depends naturally on which system we choose. Suppose we want to formulate physical laws within a system of an m-dimensional spacetime embedded in  N-dimensional ambient manifold, we get, then, physical laws of a system having (N-m) extra dimensions. The directions of these dimensions determine those of the spacetime’s thicknesses pointing outwards away from it.

We can also describe the same physical laws in a much simpler system where the same N-ambient space embedding (N-1)-hypersurface, instead of an m-spacetime. The thickness of such a hypersurface has the same direction as at the Nth dimension pointing outward away from it.

The laws of nature in the first system have very complex formulations and are difficult to resolve, as the system has too many extra-dimensions and, hence, fewer symmetries.  The laws of nature in the second system are relatively more straightforward as the system has only one extra-dimension and is highly symmetric.

We can best describe the laws of nature when the number of the dimensions of the ambient space embedding the system is large enough which "stretches" out the hypersurface to become completely flat and perfectly symmetric.

How do we determine the dimensions of the ambient space (N) vis-a-vis that of the embedded spacetime (m)? There is a minimum requirement for the number of the ambient space's dimensions in order that the spacetime can be “properly" embedded in the ambient space. The [non-flat] m-spacetime can be embedded in N-manifold only if at least N = ½ m(m+1) 1). The metric tensor of the m-spacetime dictates that the ambient space should have that amount of dimensions for all of its components can be properly defined.

Based on the above rule, the 2-surface requires  3-ambient space for which we do not doubt it. The non-flat 3-space, in our surprise,  requires 6-ambient spacetime, not to mention the 4-spacetime which requires 10-ambient manifold. It may indirectly explain why we have three generations of elementary particles and the 10-ambient manifold as revealed in the current theoretical physics. 

Multidimensional Time

Now, what are these dimensions all about? As we have discussed previously, the spacetime is the physical manifestation of energy. In its original state, the spacetime was perfectly symmetric. All of its dimensions are indistinguishable, and they are all "temporal." When the respective energy segregates into the positive and negative energies, the [temporal] spacetime's dimensions along the interface [separating those opposing energies] are transformed into spatial dimensions.  

For the classical 4-spacetime, the energy segregation transforms three of the spacetime's temporal dimensions along the interface into spatial (Figure-3). In a 6-spacetime, the energy’s segregation transforms the spacetime's five temporal dimensions along the interface into spatial dimensions. The same case also prevails for the 10-spacetime., where nine temporal dimensions along the interface become spatial.
The temporal dimensions  t1, t3 and t7 related to the 4-, 6- and 10-spacetimes, respectively, are different from each other. It is against the mainstream premise which tacitly asserts that there is only one temporal dimension in nature.
Based on the rule we have, a 4-spacetime requires a 10-ambient space for the physical laws to have solutions. However, as we have in this case 3 spatial dimensions and seven [imaginary] extra-temporal dimensions, the physical laws we get would be very complicated. It is imperative, therefore, to have the same laws applied to a system consisting of a 10-ambient space embedding 9-hypersurface, which are simpler as we have only one imaginary temporal dimension on top of the nine real ones.

 It is more or less what physicists have done in developing the string theory, except that the extra-dimensions were assumed to curl into tiny loops. Also, the temporal dimension of the system was assumed to be the same as that of ordinary time. Such wrong assumptions have been put forward because mainstream physics holds the premise that time is one-dimensional as previously mentioned. 


The relativity theory should rigorously hold the equivalence of space and time dimensions. The spatial and temporal dimensions should be transferable to each other depending on the system they become part. The extra dimensions are undetectable not because they curl into tiny loops but because they are temporal.


Supermanifold and Supersymmetry Generators

Physicists have many problems with their mathematical propositions as they used to conceptualize the spacetime as a standalone basis. Under such a concept they have taken the part of the reality out of the system. Such as is the case of the Big Bang theory, which is entirely Platonic, a system without any geometrical thicknesses, surrounding, nor even 3-space.

A reader of the Scientific American2) once asked: "Where is the universe expanding to?"  The authoritative answer from the expert was: "... the universe's expansion does not push it into new territory - rather the spacetime grid itself is expanding".  The issue has arisen again and again since the Big Bang theory was put forward, as only a few people were satisfied with such an explanation. The excellent answer should be that the universe is expanding to at least the 10-dimensional ambient space, and not into nothing.
To make their model closer to the reality, some physicists artificially introduced what they called supersymmetry generators, replacing the thicknesses which they have “forgotten” to incorporate in their mathematical model. They call this manifold having thicknesses “Supermanifold”3). The physicists should put forward the problems of embedding at the forefront of physical researches and develop a more holistic model instead of a piecemeal one.

References:
1.      Sokolnikoff, L.S.: ”Tensor Analysis," Wiley Toppan, Second Edition, New York, 1964, p. 205
2.      Kashlinsky, A.: "Where is the Universe Expanding to?", Scientific American, (Ask the Experts Forum), May 2007, p. 104
3.      Penrose R.: "The Road to Reality," Vintage Books, London, 2005, p. 879